Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userCountable Tightness and the Grothendieck Property in $C_p$-Theory
55
0
0.0
(
0
)
Authors
Franklin D. Tall
Ask ChatGPT about the research
No Arabic abstract
The Grothendieck property has become important in research on the definability of pathological Banach spaces [CI], [HT], and especially [HT20]. We here answer a question of Arhangelskiu{i} by proving it undecidable whether countably tight spaces with Lindelof finite powers are Grothendieck. We answer another of his questions by proving that $mathrm{PFA}$ implies Lindelof countably tight spaces are Grothendieck. We also prove that various other consequences of $mathrm{MA}_{omega_1}$ and $mathrm{PFA}$ considered by Arhangelskiu{i}, Okunev, and Reznichenko are not theorems of $mathrm{ZFC}$.
rate research
Read More
A topological gyrogroup is a gyrogroup endowed with a topology such that the binary operation is jointly continuous and the inverse mapping is also continuous. In this paper, it is proved that if $G$ is a sequential topological gyrogroup with an $omega^{omega}$-base, then $G$ has the strong Pytkeev property. Moreover, some equivalent conditions about $omega^{omega}$-base and strong Pytkeev property are given in Baire topological gyrogroups. Finally, it is shown that if $G$ is a strongly countably complete strongly topological gyrogroup, then $G$ contains a closed, countably compact, admissible subgyrogroup $P$ such that the quotient space $G/P$ is metrizable and the canonical homomorphism $pi :Grightarrow G/P$ is closed.
A Hausdorff topological space $X$ is called $textit{superconnected}$ (resp. $textit{coregular}$) if for any nonempty open sets $U_1,dots U_nsubseteq X$, the intersection of their closures $bar U_1capdotscapbar U_n$ is not empty (resp. the complement $Xsetminus (bar U_1capdotscapbar U_n)$ is a regular topological space). A canonical example of a coregular superconnected space is the projective space $mathbb Qmathsf P^infty$ of the topological vector space $mathbb Q^{<omega}={(x_n)_{ninomega}in mathbb Q^{omega}:|{ninomega:x_n e 0}|<omega}$ over the field of rationals $mathbb Q$. The space $mathbb Qmathsf P^infty$ is the quotient space of $mathbb Q^{<omega}setminus{0}^omega$ by the equivalence relation $xsim y$ iff $mathbb Q{cdot}x=mathbb Q{cdot}y$. We prove that every countable second-countable coregular space is homeomorphic to a subspace of $mathbb Qmathsf P^infty$, and a topological space $X$ is homeomorphic to $mathbb Qmathsf P^infty$ if and only if $X$ is countable, second-countable, and admits a decreasing sequence of closed sets $(X_n)_{ninomega}$ such that (i) $X_0=X$, $bigcap_{ninomega}X_n=emptyset$, (ii) for every $ninomega$ and a nonempty open set $Usubseteq X_n$ the closure $bar U$ contains some set $X_m$, and (iii) for every $ninomega$ the complement $Xsetminus X_n$ is a regular topological space. Using this topological characterization of $mathbb Qmathsf P^infty$ we find topological copies of the space $mathbb Qmathsf P^infty$ among quotient spaces, orbit spaces of group actions, and projective spaces of topological vector spaces over countable topological fields.
It is proved that any countable topological group in which the filter of neighborhoods of the identity element is not rapid contains a discrete set with precisely one nonisolated point. This gives a negative answer to Protasovs question on the existence in ZFC of a countable nondiscrete group in which all discrete subsets are closed. It is also proved that the existence of a countable nondiscrete extremally disconnected group implies the existence of a rapid ultrafilter and, hence, a countable nondiscrete extremally disconnected group cannot be constructed in ZFC.
As proved in [16], for a Tychonoff space $X$, a locally convex space $C_{p}(X)$ is distinguished if and only if $X$ is a $Delta$-space. If there exists a linear continuous surjective mapping $T:C_p(X) to C_p(Y)$ and $C_p(X)$ is distinguished, then $C_p(Y)$ also is distinguished [17]. Firstly, in this paper we explore the following question: Under which conditions the operator $T:C_p(X) to C_p(Y)$ above is open? Secondly, we devote a special attention to concrete distinguished spaces $C_p([1,alpha])$, where $alpha$ is a countable ordinal number. A complete characterization of all $Y$ which admit a linear continuous surjective mapping $T:C_p([1,alpha]) to C_p(Y)$ is given. We also observe that for every countable ordinal $alpha$ all closed linear subspaces of $C_p([1,alpha])$ are distinguished, thereby answering an open question posed in [17]. Using some properties of $Delta$-spaces we prove that a linear continuous surjection $T:C_p(X) to C_k(X)_w$, where $C_k(X)_w$ denotes the Banach space $C(X)$ endowed with its weak topology, does not exist for every infinite metrizable compact $C$-space $X$ (in particular, for every infinite compact $X subset mathbb{R}^n$).
We prove that the locally convex space $C_{p}(X)$ of continuous real-valued functions on a Tychonoff space $X$ equipped with the topology of pointwise convergence is distinguished if and only if $X$ is a $Delta$-space in the sense of cite {Knight}. As an application of this characterization theorem we obtain the following results: 1) If $X$ is a v{C}ech-complete (in particular, compact) space such that $C_p(X)$ is distinguished, then $X$ is scattered. 2) For every separable compact space of the Isbell--Mrowka type $X$, the space $C_p(X)$ is distinguished. 3) If $X$ is the compact space of ordinals $[0,omega_1]$, then $C_p(X)$ is not distinguished. We observe that the existence of an uncountable separable metrizable space $X$ such that $C_p(X)$ is distinguished, is independent of ZFC. We explore also the question to which extent the class of $Delta$-spaces is invariant under basic topological operations.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
